Finite Group Databasebeta

← #13175324

Subgroups — #13175324

Order 1728.

1708 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723916C211yesmaximal, characteristic, complemented
2C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#870C211yesmaximal, characteristic, complemented
4#11723526C211yesmaximal, characteristic
5#11723512C212yesmaximal, complemented
6#11723494C211yesmaximal, characteristic, complemented
7#11723469C212yesmaximal, complemented
8#11723446C212yesmaximal, complemented
9#11723431C211yesmaximal, characteristic, complemented
10#11723417C212yesmaximal, complemented
11#11723406C211yesmaximal, characteristic, complemented
12#10596162—31nomaximal
13#10593976—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×#6292C2212yes—
20C2×#6289C2212yes—
21C33×C42C2211yescharacteristic
22#88859C2212yes—
23#88859C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30C32×Dic3⋊C4C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#125—31no—
38#64107—32no—
39#64106—31no—
40#64104—31no—
41#64103—32no—
42#64102—32no—
43#64102—31no—
44#64101—32no—
45#64096—31no—
46#64084—31no—
47#64084—34no—
48#64084—64no—
49C42×C3⋊D3D311yescharacteristic, complemented
50C3×C42×D3—34no—
51C3×#870D314yescomplemented
52#63909—34no—
53#63904—38no—
54#63894—34no—
55#63868—38no—
56#63857—34no—
57#63850—38no—
58#63844—34no—
59#63836—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊(C3⋊D3))—22no—
67#6308—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4×C3⋊D3—22no—
72C3×C4×C3⋊D3—22no—
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2312yes—
76#6292—24no—
77#6289—24no—
78#5691—94no—
79#5479—91no—
80C2×C4⋊(C3⋊D3)—32no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4⋊(C3⋊D3)—34no—
83C2×C4⋊(C3⋊D3)—64no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×#634—31no—
89C2×#634—34no—
90C2×#634—64no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—34no—
93C2×C3×C4×D3—34no—
94C2×#626—38no—
95C2×#624—38no—
96C32×C4⋊C4—31no—
97C32×C4⋊C4—32no—
98C32×C4⋊C4—31no—
99C32×C4⋊C4—32no—
100C32×C42—24no—
101C32×C42C2×D314yes—
102C32×C42C2×D311yescharacteristic
103#3752—32no—
104#3752—32no—
105#3752—32no—
106#3752—38no—
107#3752—38no—
108#3752—32no—
109#3752—32no—
110#3752—68no—
111#3752—68no—
112C4⋊(C3⋊Dic3)—32no—
113#3750—32no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C4×C3⋊Dic3C2×D311yescharacteristic
117C4×C3⋊Dic3C2×D311yescharacteristic
118C3×C4⋊Dic3C2×D314yes—
119C3×C4⋊Dic3C2×D314yes—
120C3×Dic3⋊C4C2×D318yes—
121C3×Dic3⋊C4C2×D318yes—
122C3×C4×Dic3—34no—
123C3×C4×Dic3—34no—
124C3×C4×Dic3—34no—
125#3724—38no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130Dic32—38no—
131C22×C33C2411yescharacteristic
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3#4112yes—
138C33×C4—21no—
139C33×C4—21no—
140C33×C4#4112yes—
141C3⋊(C3⋊Dic3)—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3#4112yes—
145C32×Dic3—22no—
146C32×Dic3—22no—
147C3×#125—34no—
148C2×C3×C42—91no—
149#930—91no—
150#929—92no—
151C4×#67—92no—
152#924—91no—
153#898—98no—
154#897—94no—
155#895—94no—
156#894—98no—
157#893—94no—
158#893—98no—
159#892—98no—
160#887—94no—
161#876—92no—
162#875—94no—
163#875—91no—
164#875—184no—
165C4×D12—91no—
166#872—92no—
167C42×D3—34no—
168#870C3⋊D311yescharacteristic, complemented
169C4×Dic6—91no—
170C22×C3⋊D3—32no—
171C22×C3⋊D3—38no—
172C22×C3⋊D3—68no—
173C22×C3⋊D3C22×D311yescharacteristic
174C22×C3×D3—34no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—32no—
179C2×C32×C4—32no—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4⋊(C3⋊D3)—64no—
193C4⋊(C3⋊D3)—62no—
194C4⋊(C3⋊D3)—68no—
195C4⋊(C3⋊D3)—128no—
196C4×C3⋊D3—22no—
197C4×C3⋊D3—22no—
198C4×C3⋊D3—22no—
199C4×C3⋊D3—64no—
200#634—62no—
201#634—68no—
202#634—128no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3C22×D318yes—
207C2×C3×Dic3C22×D318yes—
208C3×C4×D3—68no—
209C3×C4×D3—68no—
210C3×C4×D3—68no—
211#626—616no—
212#624—616no—
213#529—271noSylow
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C3⋊(C3⋊D3)—42no—
218C3×C3⋊D3—22no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C2×#67—92no—
223C2×D12—98no—
224C2×D12—94no—
225C2×D12—91no—
226C2×D12—184no—
227C2×C4×D3—34no—
228C2×C4×D3—34no—
229C2×C4×D3—34no—
230C2×C4×D3—98no—
231C2×C4×D3—92no—
232C2×Dic6—94no—
233C2×Dic6—91no—
234C2×Dic6—184no—
235C3×C4⋊C4—34no—
236C3×C4⋊C4—38no—
237C3×C4⋊C4—34no—
238C3×C4⋊C4—38no—
239C3×C42—24no—
240C3×C42—91no—
241C3×C42—91no—
242C3×C42—91no—
243C3×C42D3214yes—
244C3×C42C2×C3⋊D311yescharacteristic
245#217—92no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—98no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—188no—
256(C2×D3)⋊C4—188no—
257C4⋊Dic3—98no—
258C4⋊Dic3C2×C3⋊D311yescharacteristic
259C4⋊Dic3C2×C3⋊D311yescharacteristic
260Dic3⋊C4—98no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4C2×C3⋊D312yes—
264Dic3⋊C4C2×C3⋊D312yes—
265C4×Dic3—34no—
266C4×Dic3—34no—
267C4×Dic3—34no—
268C4×Dic3—98no—
269C4×Dic3—92no—
270C22×C32—24no—
271C22×C32C23×D314yes—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—21no—
274C2×C3⋊D3—21no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—68no—
281C2×C3⋊D3—128no—
282C2×C3⋊D3—128no—
283C2×C3⋊D3—128no—
284C2×C3⋊D3#23512yes—
285C2×C3×D3—38no—
286C2×C3×D3—64no—
287C2×C3×D3—64no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—28no—
291C32×C4—24no—
292C32×C4—24no—
293C32×C4—44no—
294C32×C4—44no—
295C32×C4—62no—
296C32×C4—62no—
297C32×C4#23512yes—
298C32×C4#23918yes—
299C3⋊Dic3—21no—
300C3⋊Dic3—21no—
301C3⋊Dic3—62no—
302C3⋊Dic3—68no—
303C3⋊Dic3—128no—
304C3⋊Dic3#23512yes—
305C3×Dic3—28no—
306C3×Dic3—28no—
307C3×Dic3—38no—
308C3×Dic3—64no—
309C3×Dic3—64no—
310#126—272no—
311#125—31no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—271no—
316C4×Q8—271no—
317C4×D4—271no—
318C4×D4—272no—
319#117—272no—
320C2×C42—91no—
321C33#52911yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C22×D3—34no—
324C22×D3—98no—
325C22×D3—92no—
326C22×D3—188no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—38no—
331C2×C3×C4—38no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—182no—
336C2×C3×C4—182no—
337C2×C3×C4—182no—
338C2×C3×C4C2×D3214yes—
339C2×C3×C4C2×D3214yes—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342C2×C3×C4C22×C3⋊D311yescharacteristic
343C2×C3×C4C22×C3⋊D311yescharacteristic
344#67—184no—
345C2×Dic3—34no—
346C2×Dic3—34no—
347C2×Dic3—34no—
348C2×Dic3—98no—
349C2×Dic3—92no—
350C2×Dic3—188no—
351C2×Dic3C22×C3⋊D312yes—
352C2×Dic3C22×C3⋊D312yes—
353D12—1816no—
354D12—188no—
355D12—182no—
356D12—368no—
357C4×D3—68no—
358C4×D3—68no—
359C4×D3—68no—
360C4×D3—184no—
361C4×D3—1816no—
362Dic6—188no—
363Dic6—182no—
364Dic6—368no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32C2×#23511yescharacteristic
369C2×C32C2×#23511yescharacteristic
370C2×C32C2×#23511yescharacteristic
371C2×C32C2×#23914yes—
372C2×C32D3×#4114yes—
373C2×C32D3×#4114yes—
374C3⋊D3—22no—
375C3⋊D3—122no—
376C3⋊D3—128no—
377C3⋊D3—248no—
378C3×D3—68no—
379C2×Q8—271no—
380C2×D4—272no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—272no—
386C4⋊C4—32no—
387C4⋊C4—31no—
388C4⋊C4—32no—
389C4⋊C4—31no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—91no—
398C42—91no—
399C42—91no—
400C42—272no—
401C42D3×C3⋊D311yescharacteristic
402C22×C3—24no—
403C22×C3—92no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3C22×D3214yes—
407C22×C3C23×C3⋊D311yescharacteristic
408C2×D3—38no—
409C2×D3—64no—
410C2×D3—64no—
411C2×D3—188no—
412C2×D3—188no—
413C2×D3—188no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—368no—
418C2×D3—368no—
419C2×D3—368no—
420C3×C4—24no—
421C3×C4—24no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—28no—
425C3×C4—44no—
426C3×C4—44no—
427C3×C4—68no—
428C3×C4—68no—
429C3×C4—92no—
430C3×C4—181no—
431C3×C4—181no—
432C3×C4#379918yes—
433C3×C4#383212yes—
434Dic3—22no—
435Dic3—22no—
436Dic3—38no—
437Dic3—64no—
438Dic3—64no—
439Dic3—188no—
440Dic3—182no—
441Dic3—368no—
442C32—24no—
443C32#547911yescharacteristic, complemented
444C32#569114yescomplemented
445C23—91no—
446C23—272no—
447Q8—542no—
448D4—544no—
449D4—542no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—182no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4—272no—
459C2×C4—544no—
460C2×C4C2×D3×C3⋊D311yescharacteristic
461C2×C4C2×D3×C3⋊D311yescharacteristic
462C2×C4C2×D3×C3⋊D311yescharacteristic
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—182no—
467C2×C3C2×#379914yes—
468C2×C3D3×#23514yes—
469C2×C3D3×#23514yes—
470C2×C3C2×#383211yescharacteristic
471C2×C3#41×C3⋊D311yescharacteristic
472C2×C3#41×C3⋊D311yescharacteristic
473D3—68no—
474D3—362no—
475D3—368no—
476D3—728no—
477C22—92no—
478C22—181no—
479C22—181no—
480C22—542no—
481C22—542no—
482C22—542no—
483C22C22×D3×C3⋊D311yescharacteristic
484C4—21no—
485C4—21no—
486C4—62no—
487C4—62no—
488C4—92no—
489C4—181no—
490C4—181no—
491C4—542no—
492C4#8901812yes—
493C3—24no—
494C3#1059397614yescomplemented
495C3#1059616211yescharacteristic, complemented
496C2—182no—
497C2—1082no—
498C2C2×#8901811yescharacteristic
499C2C3⋊D3×#23511yescharacteristic
500C2C3⋊D3×#23511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723526C2
hypercenterC42D3×C3⋊D3
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11723916C2semidirect
2. C3×C42×C3⋊D3C2semidirect
3. C32×#870C2semidirect
5. #11723512C2semidirect
6. #11723494C2semidirect
7. #11723469C2semidirect
8. #11723446C2semidirect
9. #11723431C2semidirect
10. #11723417C2semidirect
11. #11723406C2semidirect
49. C42×C3⋊D3D3semidirect
51. C3×#870D3semidirect
168. #870C3⋊D3semidirect
321. C33#529semidirect
443. C32#5479semidirect
444. C32#5691semidirect
494. C3#10593976semidirect
495. C3#10596162semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117239162
C3×C42×C3⋊D32
C32×#8702
#117235262
#117235122
#117234942
#117234692
#117234462
#117234312
#117234172
#117234062
#105961623
#105939763